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Allen G. Precalculus 1 year, 5 months ago
You can put this solution on YOUR website! a box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides a) find a function that models the volume of the box Draw the picture of the rectangular piece Sketch the square corners that are cut out of the rectangle. Each square has dimensions x by x. Imagine folding up the sides to form a box. ----------- The base of the box is (12-2x)by(20-2x) in area The height of the box is x The Volume of the box is x(12-2x)(20-2x) V = x(240-24x-40x+4x^2) V = 4x^3 - 64x^2 + 240x ------------------------------------- b) find the values of x for which the volume is greater than 200 in^3 Solve 4x^3 - 64x^2 + 240x - 200 > 0 I graphed it and found the x value is greater than 10.928... But that x-value is not possible since one of the base dimensions is 20-2x. So no x value gives a volume greater than 200 in^3 ------- c) find the largest volume that such a box can have. I'll leave that to you. Cheers, Stan H. Solution:From the diagram, we know that x is also the height of the box. Then, the length, L=20-2xand the width, W=12-2x. The volume of the box, V=LWx, meaning that V=(20-2x)(12-2x)(x). Expanding the brackets and simplifying leads us to, V=4x^{3}-64x^{2}+240xTo figure out the domain, the following conditions must be true. The length, L>0\Leftrightarrow 20-2x>0\Leftrightarrow x<10and the width, W>0\Leftrightarrow 12-2x>0\Leftrightarrow x<6, x>0. Combining these leads us to, 0<x<6 See a mistake? Comment below so we can fix it! |
A box with an open top is to be constructed from a square piece of cardboard with dimensions 12 in
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